The Mathematics Of Factoring Teaching Tips: Challenges And .
Developmental Math – An Open CurriculumInstructor GuideUnit 12 – Table of ContentsUnit 12: Introduction to FactoringLearning Objectives12.2Instructor Notes12.3 The Mathematics of FactoringTeaching Tips: Challenges and ApproachesAdditional ResourcesInstructor Overview 12.10Tutor Simulation: Playing the Elimination GameInstructor Overview 12.11Puzzle: Match FactorsInstructor Overview 12.13Project: Making ConnectionsCommon Core Standards12.43Some rights reserved. See our complete Terms of Use.Monterey Institute for Technology and Education (MITE) 2012To see these and all other available Instructor Resources, visit the NROC Network.12.1
Developmental Math – An Open CurriculumInstructor GuideUnit 12 – Learning ObjectivesUnit 12: FactoringLesson 1: Introduction to FactoringTopic 1: Greatest Common FactorLearning Objectives Find the greatest common factor (GCF) of monomials. Factor polynomials by factoring out the greatest common factor (GCF). Factor expressions with four terms by grouping.Lesson 2: Factoring PolynomialsTopic 1: Factoring TrinomialsLearning Objectives Factor trinomials with a leading coefficient of 1. Factor trinomials with a common factor. Factor trinomials with a leading coefficient other than 1.Topic 2: Factoring: Special CasesLearning Objectives Factor trinomials that are perfect squares. Factor binomials in the form of the difference of squares.Topic 3: Special Cases: CubesLearning Objectives Factor the sum of cubes. Factor the difference of cubes.Lesson 3: Solving Quadratic EquationsTopic 1: Solve Quadratic Equations by FactoringLearning Objectives Solve equations in factored form by using the Principle of Zero Products. Solve quadratic equations by factoring and then using the Principle of Zero Products. Solve application problems involving quadratic equations.12.2
Developmental Math – An Open CurriculumInstructor GuideUnit 12 – Instructor NotesUnit 12: FactoringInstructor NotesThe Mathematics of FactoringThis unit builds upon students’ knowledge of polynomials learned in the previous unit. They willlearn how to use the distributive property and greatest common factors to find the factored formof binomials and how to factor trinomials by grouping. Students will also learn how to recognizeand quickly factor special products (perfect square trinomials, difference of squares, and thesum and difference of two squares). Finally, they’ll get experience combining these techniquesand using them to solve quadratic equations.Teaching Tips: Challenges and ApproachesThis unit on factoring is probably one of the most difficult—students will spend a lot of timecarrying out multi-step, complex procedures for what will often seem to be obscure purposes.At this stage in algebra, factoring polynomials may feel like busy work rather than a means to auseful end. It doesn’t help that students may remember having trouble with factoring from whenthey studied algebra in high school.Encourage students to think of factoring as the reverse of multiplying polynomials that waslearned previously. Then, a problem multiplying polynomials was given and students wereasked to calculate the answer. In this unit, the answer is given and the students need to comeup with the question. Sound familiar? In a way, factoring is like playing the popular game showJeopardy.Greatest Common FactorFinding the greatest common factor of whole numbers should be reviewed before finding theGCF of polynomials. Then it is a logical step to demonstrate how to factor expressions by usingthe distributive property in reverse to pull out the greatest common monomial from each term ina polynomial:12.3
Developmental Math – An Open CurriculumInstructor Guide[From Lesson 1, Topic 1, Topic Text]Remind your students to pay particular attention to signs as it is easy to make a mistake withthem, and also to check their final answers by multiplying.GroupingAfter your students are comfortable pulling the GCF out of a polynomial, it is time to teach thema new method of factoring–factoring by grouping. Begin by introducing the technique on 4-termpolynomials. It's fairly easy for students to understand how to break these polynomials intogroups of two and then factor each pair.Trinomials are trickier. Indeed, many textbooks do not use grouping for factoring trinomials, andinstead use essentially a guess and check method. While factoring by grouping may initially bea more complex procedure, it has many significant advantages in the long term and is used inthis course. The hardest part is figuring out how to rewrite the middle term of a trinomial as anequivalent binomial. Students will need to see this demonstrated repeatedly, as well as get a lotof practice working on their own. Even after they grasp the basic idea, they'll often have troubledeciding which signs to use. It will be helpful to supply them with a set of tips like the one below:12.4
Developmental Math – An Open CurriculumInstructor Guide[From Lesson 2, Topic 1, Topic Text]Factoring by grouping has the great advantage of working for all trinomials. It also provides amethod to determine when a polynomial cannot be factored. (This is not obvious when studentsare using the guess and check method.)Sometimes students don’t remember to look for the greatest common factor of all the terms of apolynomial before trying to factor by grouping. This isn’t wrong, but the larger numbers canmake the work more difficult. Plus the student has to remember to look for a greatest commonfactor at the end anyway. In order to illustrate this, have students factor 9x2 15x 36 withoutpulling out the greatest common factor of 3 -- they will notice that the numbers are cumbersome.After this, have them try again, this time factoring out the 3 as the first step. They will see thebenefits.Once the grouping method is mastered, let your students use it to factor perfect squaretrinomials. Hopefully they'll soon see a pattern, though you will probably have to nudge themalong. Eventually, they should learn to recognize if a trinomial is a perfect square, and be ableto factor it without grouping.After the rule for factoring a perfect square trinomial has been developed, set them to findingone for factoring the difference of two squares. This rule is usually very easy for students tofigure out. Then have them try to factor the sum of two squares, such as x2 4. Make surethey understand that this cannot be done.12.5
Developmental Math – An Open CurriculumInstructor GuideIntermediate algebra students will also need to know how to factor the sum and difference oftwo cubes. They are sure to have trouble remembering the formulas. Try pointing out that theformulas are really the same except for signs: A binomial in the form a3 b3 can be factored as (a b)(a2 – ab b2)A binomial in the form a3 – b3 can be factored as (a – b)(a2 ab b2)The sign in between the two cubes is the same sign as in the first factor in the formulas. Thenext sign is the opposite of the first sign and the last sign is always positive. Now “all” they haveto remember are the variable parts of the formulas. Easy!Factoring Quadratic EquationsThe last topic in this unit is solving quadratic equations by factoring and applying the zeroproducts rule. Begin by solving an example where the polynomial is already factored and setequal to zero, such as the following:[From Lesson 3, Topic 1, Worked Example 1]Now give your students a problem like "Solve x2 x – 12 0 for x." Ask them how they wouldattempt to solve for x. Someone will suggest factoring the left hand side by grouping and theywill be on their way.Then pose the problem x2 x – 12 18. Make sure your students know that in order for theprinciple of zero products to work, the trinomial must be set equal to 0. Sometimes students are12.6
Developmental Math – An Open CurriculumInstructor Guideso focused on new techniques, they forget basic principles for rewriting an equation and theymay need to be prodded to add (or subtract) something to (or from) both sides so that one sideequals zero.Be careful -- once students get into the hang of applying the zero products rule to solveequations, they may start trying it on expressions as well. For instance, if a problem says tofactor x2 x – 12, some will do so and then go ahead and calculate that x -4 or 3. Remindyour students to only do what a problem asks – factor when it says to factor and solve when itsays to solve.The Sense TestApplication problems have an extra requirement that solving given equations do not -- answershave to make sense based on their context. Consider the following scenario:[From Lesson 3, Topic 1, Topic Text]12.7
Developmental Math – An Open CurriculumInstructor GuideMathematically, it is true that t can be either 4 or 1. But logically, only one of these answers2works -- since t represents the number of seconds after the rocket has taken off, it can’t be anegative number. The rocket can't hit the ground before it was launched. Teach students thatwhen they do application problems like this, they need to check not only the math but also thesense of their results.Keep in MindFactoring trinomials and solving quadratic equations are difficult topics. As soon as you say“factoring,” some students will recall hours of erasing before correct answers were foundthrough trial and error. Reassure students that while the factoring by grouping method takeslonger to use when working simple problems, it really will make solving complex problemsquicker. Stress to your students that once something is factored, they should check theirwork by multiplying. This will help them catch any errors that were made.Most of the material in this unit has been geared to both beginning and intermediate students.More difficult examples and problems are included for the intermediate algebra student, butthese could be used to challenge the beginning algebra student. However, two topics, factoringthe sum and difference of two cubes, are intended only for intermediate algebra students.Additional ResourcesIn all mathematics, the best way to really learn new skills and ideas is repetition. Problemsolving is woven into every aspect of this course—each topic includes warm-up, practice, andreview problems for students to solve on their own. The presentations, worked examples, andtopic texts demonstrate how to tackle even more problems. But practice makes perfect, andsome students will benefit from additional work.Practice finding the common factor of polynomials at http://www.mathsnet.net/algebra/a41.html(get additional problems by clicking on “more on this topic”).Factoring practice using the AC Method can be found ra/AC/AC.html.Solve quadratic equations using the principle of zero products athttp://www.mathsnet.net/algebra/e34.html (get additional problems by clicking on “more on thistopic”).Practice all types of factoring problems /02/pageGenerate?site quickmath&s1 algebra&s2 factor&s3 basic.Review factoring and solving quadratic equations at http://www.quia.com/rr/36611.html.12.8
Developmental Math – An Open CurriculumInstructor GuideSummaryAfter completing this unit, students will be more comfortable with factoring any polynomial that isgiven to them. They'll be able to pull out the GCF and factor by grouping, and recognize specialcases such as perfect square trinomials, difference of two squares, and the sum and differenceof two cubes. They'll have had experience combining these techniques to solve quadraticequations, and will have gained an appreciation that factoring can be used to solve real-lifeproblems.12.9
Developmental Math – An Open CurriculumInstructor GuideUnit 12 – Tutor SimulationUnit 12: FactoringInstructor OverviewTutor Simulation: Playing the Elimination GamePurposeThis simulation allows students to demonstrate their ability to factor polynomials. Students willbe asked to apply what they have learned to solve a problem involving: Factoring out the greatest common factorFactoring by groupingFactoring the sum or difference of perfect squaresFactoring the sum or difference of cubesProblemStudents are presented with the following problem:Many mathematicians have tricks they use to analyze an expression in order to determine ifthey can factor it quickly. In this simulation, we are going to focus on techniques for quicklyfactoring polynomials with two terms (binomials) or with four terms.RecommendationsTutor simulations are designed to give students a chance to assess their understanding of unitmaterial in a personal, risk-free situation. Before directing students to the simulation, Make sure they have completed all other unit material.Explain the mechanics of tutor simulations.o Students will be given a problem and then guided through its solution by a videotutor;o After each answer is chosen, students should wait for tutor feedback beforecontinuing;o After the simulation is completed, students will be given an assessment of theirefforts. If areas of concern are found, the students should review unit materials orseek help from their instructor.Emphasize that this is an exploration, not an exam.12.10
Developmental Math – An Open CurriculumInstructor GuideUnit 12 – PuzzleUnit 12: FactoringInstructor OverviewPuzzle: Match FactorsObjectivesMatch Factors is a puzzle that tests a player's ability to factor by grouping. It reinforces thetechnique of factoring a trinomial in the form ax2 bx c by finding two integers, r and s, whosesum is b and whose product is ac. Puzzle play, especially when done by eye rather than withpencil and paper, will help students learn to quickly identify the components of factors.Figure 1. Match Factors players choose the factors of a central polynomial from a rotating ring ofpossibilities.DescriptionEach Match Factors game consists of a sequence of 4 polynomials surrounded by 8 possiblefactors. As each polynomial is displayed, players are asked to pick the matching pair of factors.If they choose correctly, the next polynomial appears. If not, they must try again before playadvances.12.11
Developmental Math – An Open CurriculumInstructor GuideThere are three levels of play, each containing 10 games. In Level 1, polynomials have the formx2 bx c. Level 2 polynomials have the form ax2 bx c. Players in Level 3 must factor ax2 bxy cy2 polynomials.Match Factors is suitable for individual or group play. It could also be used in a classroomsetting, with the whole group taking turns calling out the two factors of each expression.12.12
Developmental Math – An Open CurriculumInstructor GuideUnit 12 – ProjectUnit 12: FactoringInstructor OverviewProject: Making ConnectionsStudent InstructionsIntroductionThe main business of science is to uncover patterns. Often we represent those patterns asalgebraic expressions, graphs, or tables of numbers (data). Being able to make connectionsamong those various representations is one of the most vital skills to possess.TaskIn this project you attempt to make precise connections among these three ways of representingpatterns.Instructions (See the online course materials for full size graphs and charts)Work with at least one other person to complete the following exercises. Solve each problem inorder and save your work along the way. You will create a presentation on one of the four partsto be given to your class. First Problem – Connecting Algebraic Expressions and Graphs: Factor each of thefollowing expressions completely, and then compare the factored form with the “picture” ofthe expression that is shown in the graph on the right. Describe any connections that yousee, and then repeat for the next expression. In the end, formulate a conjecture as to howan algebraic expression in factored form is related to its corresponding graph. Keep in mindthat we are not assuming that you have any knowledge whatsoever about graphs. That iswhat makes this “detective work” so fun!12.13
Developmental Math – An Open CurriculumInstructor GuideAlgebraicExpressionGraph2a3Factored Form3b 12Factored Form6x2 54Factored Form12.14PossibleRelationship
Developmental Math – An Open CurriculumInstructor GuideAlgebraicExpressionGraphx2 6x 9Factored Formy2 10y 25Factored Form2c 2 4c 6Factored Form12.15PossibleRelationship
Developmental Math – An Open CurriculumInstructor GuideAlgebraicExpressionGraph3a3 24Factored Form3a3 24Factored Form2x2 4x 30Factored Form12.16PossibleRelationship
Developmental Math – An Open CurriculumInstructor GuideAlgebraicExpressionGraph5d4 15d3Factored Form2x5 4x 4 30x3Factored Form12.17PossibleRelationship
Developmental Math – An Open CurriculumInstructor GuideThe following problems are a special challenge. You may have to adjust the relationshipyou expressed above to accommodate these new examples.2x2 5x 3Factored Form6t 2 t 2Factored Form Second Problem – Connecting Algebraic Expressions and Tables: For each of thesame algebraic expressions that you examined above, compare the factored form with thetable of values associated with the expression. (For example, if the expression is2x2 5x 3 , then the value associated with it when x 1 will be 2(1)2 5(1) 3 4 .) Inthe end, formulate a conjecture that describes how an algebraic expression in factored formis related to its corresponding data table.12.18
Developmental Math – An Open CurriculumInstructor GuideAlgebraicExpression2a3Factored Form3b 12TablePossible -432-250-128-54-16-20216541282504326861024b3b 12-8-36-7-33-6-3012.19
Developmental Math – An Open CurriculumInstructor GuideAlgebraicExpressionFactored Form6x2 54TablePossible 40536679812x6x2 54-8330-7240-616212.20
Developmental Math – An Open CurriculumInstructor GuideAlgebraicExpressionFactored Formx2 6x 9TablePossible 596616272408330xx2 6x 9-825-716-6912.21
Developmental Math – An Open CurriculumInstructor GuideAlgebraicExpressionFactored Formy2 10y 25TablePossible 08121yy2 10y 25-8169-7144-612112.22
Developmental Math – An Open CurriculumInstructor GuideAlgebraicExpressionFactored Form2c 2 4c 6TablePossible 489c2c 2 4c 6-890-764-64212.23
Developmental Math – An Open CurriculumInstructor GuideAlgebraicExpressionFactored Form3a3 24TablePossible 071208154a3a3 24-8-1560-7-1053-6-67212.24
Developmental Math – An Open CurriculumInstructor GuideAlgebraicExpressionFactored Form3a3 24TablePossible 3574168535166247100581512a3a3 24-8-1512-7-1005-6-62412.25
Developmental Math – An Open CurriculumInstructor GuideAlgebraicExpressionFactored Form2x2 4x 30TablePossible 216539966727105381560x2x2 4x 30-8130-796-66612.26
Developmental Math – An Open CurriculumInstructor GuideAlgebraicExpressionFactored Form5d4 15d3TablePossible -1450618740866d5d4 15d3-828160-717150-6972012.27
Developmental Math – An Open CurriculumInstructor GuideAlgebraicExpressionTableFactored Form2x 4x 30x543Possible 4320512506324076860812800x2x5 4x 4 30x3-8-66560-7-3292812.28
Developmental Math – An Open CurriculumInstructor GuideAlgebraicExpressionFactored Form2x2 5x 3TablePossible 322-2403-6484-8965063888713720833792x2x2 5x 3-885-760-63912.29
Developmental Math – An Open CurriculumInstructor GuideAlgebraicExpressionFactored Form6t 2 t 2TablePossible 71308165t6t 2 t 2-8390-7299-622012.30
Developmental Math – An Open CurriculumInstructor GuideAlgebraicExpressionFactored Form TablePossible 620872858374Third Problem – Applying Your Findings: For each expression, factor it completely andwrite the factored form beneath the expression. Then match it to its corresponding table orgraph by writing the letter corresponding to the expression on its matching table or graph.12.31
Developmental Math – An Open CurriculumInstructor Guidea) 3c 12b) 2a5 9a4c) 3d5 12d3d) y3 27e) 2z3 16f) 2x2 5x-12g) 3a7h) t 2 6t 7i) 9y2 6y 112.32
Developmental Math – An Open CurriculumInstructor 74630589216012.33
Developmental Math – An Open CurriculumInstructor 98086416724706297670862914568100812.34
Developmental Math – An Open CurriculumInstructor Guide -18-1-150-121-92-63-340536679812Fourth Problem – Predicting the Unknown: One of the primary reasons to makeconnections is to be able to explain or predict previously unobserved behavior. Below weprovide you with some tables and some graphs. Based on these alone, determine whetherthe expression associated with them can be factored. Explain the reasoning behind yourdecision. [Hint: You should make use of your observations from the problems above todetermine what it means for an expression to not be factorable.]12.35
Developmental Math – An Open CurriculumInstructor 48-4-82-3-40-2-16-1-402182203444865152624873808554
Developmental Math – An Open CurriculumInstructor 7-18091622314853268471768325ConclusionsWith those from another group, compare your answers and your way of talking about theconnections between the factored form of the expressions and the graphs and tables. Work tomake sure that your explanation is clear and concise.Prepare a presentation which:1. Explains the connection between the factored expression and the corresponding graphs andtables.12.37
Developmental Math – An Open CurriculumInstructor Guide2. Describes briefly how you determined this connection (you may want to discuss some ofyour original ideas and how you needed to refine them as you looked at more examples).3. Gives a test for determining whether a given expression can be factored if you are given agraph or table associated with the expression.Finally, present your solution to your instructor.Instructor NotesWe would stress that nothing in this project assumes that students have any familiaritywhatsoever with graphing. In fact, it is precisely because they do not have this familiarity thatwe can explore this topic. The project is more about developing students’ abilities to noticeconnections between the various functional representations before they even understand howthese representations work. So, this project can serve both as a culminating project for Unit 12on factoring as well as very initial preparation for a unit on graphing.Assignment ProceduresProblem 1The relationship that they should be identifying is that the values of the variable that make eachfactor zero will correspond to the places where the graph crosses the horizontal axis. For thefirst eleven graphs, the student may very well identify the “the negative of the number in thefactor” as the place where the graph crosses the axis. For example, for2c 2 4c 6 2 (c 3) (c-1) , the graph crosses at c -3 and c 1. However, when theyencounter the last two, for example 2x2 5x 3 (2x-1) (x 3) , the graph does indeed crossat x -3 , but it crosses a second time at x 1and not at x 1, even though x 1 is the2“negative of the number that appears in the factor.” It may be a challenge for them to determinethe true connection, although the fact that they have had experience solving quadratic equationsby factoring should facilitate the process.Problem 2The connection is that the value of the variable which makes each factor equal to zero (andtherefore the entire expression equal to zero as well) is the one which corresponds to a zerovalue for the expression in the table. Notice that, since the values of the variable rise inincrements of one, the exact values of the variable that correspond to a zero value for theexpression do not actually appear in the last two tables. In these examples, the students willhave to notice that the value of the expression changes sign and, therefore, must have beenzero somewhere in between.Problem 3The following table shows the correct matching.12.38
Developmental Math – An Open CurriculumInstructor GuideGraph: fGraph: bTable: cGraph: hGraph: dGraph: iTable: gTable: eTable: aProblem 4The table below shows the solutions.Graph: not factorable since the graph Table: factorable because somewheredoes not cross the horizontal axis.between x -1 and x 0 it is equal to zero.Graph: not factorable since the graph Graph: factorable since the graphdoes not cross the horizontal axis.crosses the horizontal axis somewherebetween 4 and 6.Table: not factorable since nowhere does it appear that the expression changes signor is zero. Note that this is only speculative since the table shows data only forvalues of x that are integers. However, the students at this stage need not beattentive to this nuance.At this stage it can be helpful to tell the students in each group that for each question, you willrandomly choose one person in the group to present the group’s answer. This providesmotivation for the group as a whole to ensure that each member has a thorough understandingof all of the topics and gives the instructor feedback on how well each individual understandsthe work that was completed.Recommendations Have students work in teams to encourage brainstorming and cooperative learning.Assign a specific timeline for completion of the project that includes milestone dates.Provide students feedback as they complete each milestone.Ensure that each member of student groups has a specific job.Technology IntegrationThis project provides abundant opportunities for technology integration, and gives students thechance to research and collaborate using online technology. The students’ instructions listseveral websites that provide information on numbering systems, game design, and graphics.12.39
Developmental Math – An Open CurriculumInstructor GuideThe following are other examples of free Internet resources that can be used to support thisproject:http://www.moodle.orgAn Open Source Course Management System (CMS), also known as a Learning ManagementSystem (LMS) or a Virtual Learning Environment (VLE). Moodle has become very popularamong educators around the world as a tool for creating online dynamic websites for eachers or http://pbworks.com/content/edu overviewAllows you to create a secure online Wiki workspace in about 60 seconds. Encourageclassroom participation with interactive Wiki pages that students can view and edit from anycomputer. Share class resources and completed student work.http://www.docs.google.comAllows students to collaborate in real-time from any computer. Google Docs provides freeaccess and storage for word processing, spreadsheets, presentations, and surveys. This isideal for group projects.http://why.openoffice.org/The leading open-source office software suite for word processing, spreadsheets,presentations, graphics, databases and more. It can read and write files from other commonoffice software packages like Microsoft Word or Excel and MacWorks. It can be downloadedand used completely free of charge for any purpose.12.40
Developmental Math – An Open CurriculumInstructor GuideRubricScoreContent 4 3 2 1 Presentation/CommunicationThe solution shows a deep understanding ofthe problem including the ability to identifythe appropriate mathematical concepts andthe information necessary for its solution.The solution completely addresses allmathematical components presented in thetask.The solution puts to use the underlyingmathematical concepts upon which the taskis designed and applies proceduresaccurately to correctly solve the problemand verify the results.Mathematically relevant observations and/orconnections are made. The solution shows that the student has abroad understanding of the problem and themajor concepts necessary for its solution.The solution addresses all of themathematical components presented in thetask.The student uses a strategy that includesmathematical procedures and somemathematical reasoning that leads to asolution of the problem.Most parts of the project are correct withonly minor mathematical errors.The solution is not complete indicating thatparts of the problem are not understood.The solution addresses some, but not all ofthe mathematical components presented inthe task.The student uses a strategy that is partiallyuseful, and demonstrates some evidence ofmathematical reasoning.Some parts of the project may be correct,but major errors are noted and the studentcould not completely carry out mathematicalprocedures.There is no solution, or the solution has norelationship to the task.No evidence of a strategy, procedure, ormathematical reasoning and/or uses astrategy that does not help solve theproblem. 12.41 There is a clear, effective explanationdetailing how the problem is solved.All of the steps are included so thatthe reader does not need to inferhow and why decisions were made.Mathematical representation isactively used as a means ofcommunicating ideas related to theso
a new method of factoring–factoring by grouping. Begin by introducing the technique on 4-term polynomials. It's fairly easy for students to understand how to break these polynomials into groups of two and then factor each pair. Trinomials are trickier. Indeed, many textbooks do not use grouping for factoring trinomials, and
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Factoring . Factoring. Factoring is the reverse process of multiplication. Factoring polynomials in algebra has similar role as factoring numbers in arithmetic. Any number can be expressed as a product of prime numbers. For example, 2 3. 6 Similarly, any
Factoring Polynomials Martin-Gay, Developmental Mathematics 2 13.1 – The Greatest Common Factor 13.2 – Factoring Trinomials of the Form x2 bx c 13.3 – Factoring Trinomials of the Form ax 2 bx c 13.4 – Factoring Trinomials of the Form x2 bx c by Grouping 13.5 – Factoring Perfect Square Trinomials and Difference of Two Squares
241 Algebraic equations can be used to solve a large variety of problems involving geometric relationships. 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring Trinomials of the Form x2 bx c 5.4 Factoring Trinomials of the Form ax2 bx c 5.5 Factoring, Solving Equations, and Problem Solving
